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Mathematics LibreTexts

3.2.1: Quasilinear Elliptic Equations

There is a large class of quasilinear equations such that the associated characteristic equation has no solution \(\chi\), \(\nabla\chi\not=0\).

Set
$$
U=\{(x,z,p):\ x\in\Omega,\ z\in\mathbb{R}^1,\ p\in\mathbb{R}\}.
$$

Definition. The quasilinear equation (3.2.1) is called elliptic if the matrix \((a^{ij}(x,z,p))\) is positive definite for each \((x,z,p)\in U\).

Assume equation  (3.2.1) is elliptic and let \(\lambda(x,z,p)\) be the minimum and \(\Lambda(x,z,p)\) the maximum of the eigenvalues of \((a^{ij})\), then
$$
0<\lambda(x,z,p)|\zeta|^2\le\sum_{i,j=1}^na^{ij}(x,z,p)\zeta_i\zeta_j\le  \Lambda(x,z,p)|\zeta|^2
$$
for all \(\zeta\in\mathbb{R}\).

Definition. Equation  (3.2.1) is called uniformly elliptic if \(\Lambda/\lambda\) is uniformly bounded in \(U\).

An important class of  elliptic equations which are not uniformly elliptic (non-uniformly elliptic) is
\begin{equation}
\label{nonuniform}\tag{3.2.1.1}
\sum_{i=1}^n\frac{\partial}{\partial x_i}\left(\frac{u_{x_i}}{\sqrt{1+|\nabla u|^2}}\right)+\mbox{lower order terms}=0.
\end{equation}
The main part is the minimal surface operator (left hand side of the minimal surface equation). The coefficients \(a^{ij}\) are
$$
a^{ij}(x,z,p)=\left(1+|p|^2\right)^{-1/2}\left(\delta_{ij}-\frac{p_ip_j}{1+|p|^2}\right),
$$
\(\delta_{ij}\) denotes the Kronecker delta symbol. It follows that
$$
\lambda=\frac{1}{\left(1+|p|^2\right)^{3/2}},\ \ \Lambda=\frac{1}{\left(1+|p|^2\right)^{1/2}}.
$$
Thus equation (\ref{nonuniform}) is not uniformly elliptic.

The behavior of solutions of uniformly elliptic equations is similar to linear elliptic equations in contrast to the behavior of solutions of non-uniformly elliptic equations.
Typical examples for non-uniformly elliptic equations are the minimal surface equation and the capillary equation.

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